Numerical Linear algebra methods for datamining

Yousef SaadDepartment of Computer Science

and Engineering

University of Minnesota

William & MaryOct. 31, 2014

Introduction: a few factoids

ä Data is growing exponentially at an “alarming” rate:

• 90% of data in world today was created in last two years

• Every day, 2.3 Million terabytes (2.3×1018 bytes) created

ä Mixed blessing: Opportunities & big challenges.

ä Trend is re-shaping & energizing many research areas ...

ä ... including my own: numerical linear algebra

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Introduction: What is data mining?

Set of methods and tools to extract meaningful informationor patterns from data. Broad area : data analysis, machinelearning, pattern recognition, information retrieval, ...

ä Tools used: linear algebra; Statistics; Graph theory; Approx-imation theory; Optimization; ...

ä This talk: brief introduction – emphasis on linear algebraviewpoint

ä + our initial work on materials.

ä Focus on “Dimension reduction methods”

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Major tool of Data Mining: Dimension reduction

ä Goal is not as much to reduce size (& cost) but to:

• Reduce noise and redundancy in data before performing atask [e.g., classification as in digit/face recognition]

• Discover important ‘features’ or ‘paramaters’

The problem: Given: X = [x1, · · · , xn] ∈ Rm×n, find a

low-dimens. representation Y = [y1, · · · , yn] ∈ Rd×n of X

ä Achieved by a mapping Φ : x ∈ Rm −→ y ∈ Rd so:

φ(xi) = yi, i = 1, · · · , n

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m

n

X

Y

x

y

i

id

n

ä Φ may be linear : yi = W>xi , i.e., Y = W>X , ..

ä ... or nonlinear (implicit).

ä Mapping Φ required to: Preserve proximity? Maximizevariance? Preserve a certain graph?

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Example: Principal Component Analysis (PCA)

In Principal Component Analysis W is computed to maxi-mize variance of projected data:

maxW∈Rm×d;W>W=I

d∑i=1

∥∥∥∥∥∥yi − 1

n

n∑j=1

yj

∥∥∥∥∥∥2

2

, yi = W>xi.

ä Leads to maximizing

Tr[W>(X − µe>)(X − µe>)>W

], µ = 1

nΣni=1xi

ä SolutionW = { dominant eigenvectors } of the covariancematrix≡ Set of left singular vectors of X = X − µe>

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Unsupervised learning

“Unsupervised learning” : meth-ods that do not exploit known labelsä Example of digits: perform a 2-Dprojectionä Images of same digit tend tocluster (more or less)ä Such 2-D representations arepopular for visualizationä Can also try to find natural clus-ters in data, e.g., in materialsä Basic clusterning technique: K-means

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5PCA − digits : 5 −− 7

567

SuperhardPhotovoltaic

Superconductors

Catalytic

Ferromagnetic

Thermo−electricMulti−ferroics

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Example: Digit images (a random sample of 30)

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2-D ’reductions’:

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6PCA − digits : 0 −− 4

01234

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0.07 0.071 0.072 0.073 0.074−0.15

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Supervised learning: classification

Problem: Given labels(say “A” and “B”) for eachitem of a given set, find amechanism to classify anunlabelled item into eitherthe “A” or the “B” class.

?

?ä Many applications.

ä Example: distinguish SPAM and non-SPAM messages

ä Can be extended to more than 2 classes.

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Supervised learning: classification

ä Best illustration: written digits recognition example

Given: a set oflabeled samples(training set), andan (unlabeled) testimage.Problem: find

label of test image

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it 9

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ä Roughly speaking: we seek dimension reduction so thatrecognition is ‘more effective’ in low-dim. space

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Basic method: K-nearest neighbors (KNN) classification

ä Idea of a voting system: getdistances between test sampleand training samples

ä Get the k nearest neighbors(here k = 8)

ä Predominant class amongthese k items is assigned to thetest sample (“∗” here)

?kk

k k

k

n

n

nn

n

nnn

k

t

t

t

tt

tt

t

t

t

k

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Supervised learning: Linear classification

Linear classifiers: Find a hy-perplane which best separatesthe data in classes A and B.

Linear

classifier

ä Note: The world in non-linear. Often this is combined withKernels – amounts to changing the inner product

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Linear classifiers and Fisher’s LDA

ä Idea for two classes: Find a hyperplane which best sepa-rates the data in classes A and B.

Linear

classifier

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A harder case:

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4Spectral Bisection (PDDP)

ä Use kernels to transform

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Projection with Kernels −− σ2 = 2.7463

Transformed data with a Gaussian Kernel

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Fisher’s Linear Discriminant Analysis (LDA)

Goal: Use label information to define a good projector, i.e.,one that can ‘discriminate’ well between given classes

ä Define “between scatter”: a measure of how well separatedtwo distinct classes are.

ä Define “within scatter”: a measure of how well clustereditems of the same class are.

ä Objective: make “between scatter” measure large and “withinscatter” small.

Idea: Find projector that maximizes the ratio of the “betweenscatter” measure over “within scatter” measure

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Define:

SB =c∑

k=1

nk(µ(k) − µ)(µ(k) − µ)T ,

SW =c∑

k=1

∑xi ∈Xk

(xi − µ(k))(xi − µ(k))T

Where:

• µ = mean (X)• µ(k) = mean (Xk)• Xk = k-th class• nk = |Xk|

H

GLOBAL CENTROID

CLUSTER CENTROIDS

H

X3

1X

µ

X2

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ä Consider 2ndmoments for a vec-tor a:

aTSBa =c∑i=1

nk |aT(µ(k) − µ)|2,

aTSWa =c∑

k=1

∑xi ∈ Xk

|aT(xi − µ(k))|2

ä aTSBa ≡ weighted variance of projected µj’s

ä aTSWa ≡ w. sum of variances of projected classes Xj’s

ä LDA projects the data so as to maxi-mize the ratio of these two numbers:

maxa

aTSBa

aTSWa

ä Optimal a = eigenvector associated with the largest eigen-value of: SBui = λiSWui .

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LDA – Extension to arbitrary dimensions

ä Criterion: maximize the ratioof two traces:

Tr [UTSBU ]

Tr [UTSWU ]

ä Constraint: UTU = I (orthogonal projector).

ä Reduced dimension data: Y = UTX.

Common viewpoint: hard to maximize, therefore ...

ä ... alternative: Solve insteadthe (‘easier’) problem:

maxUTSWU=I

Tr [UTSBU ]

ä Solution: largest eigenvectors of SBui = λiSWui .

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LDA – Extension to arbitrary dimensions (cont.)

ä Consider the originalproblem:

maxU ∈ Rn×p, UTU=I

Tr [UTAU ]

Tr [UTBU ]

Let A,B be symmetric & assume that B is semi-positivedefinite with rank(B) > n− p. Then Tr [UTAU ]/Tr [UTBU ]has a finite maximum value ρ∗. The maximum is reached for acertain U∗ that is unique up to unitary transforms of columns.

ä Considerthe function:

f(ρ) = maxV TV=I

Tr [V T(A− ρB)V ]

ä Call V (ρ) the maximizer for an arbitrary given ρ.

ä Note: V (ρ) = Set of eigenvectors - not unique

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ä Define G(ρ) ≡ A− ρB and its n eigenvalues:

µ1(ρ) ≥ µ2(ρ) ≥ · · · ≥ µn(ρ) .

ä Clearly:

f(ρ) = µ1(ρ) + µ2(ρ) + · · ·+ µp(ρ) .

ä Can express this differently. Define eigenprojector:

P (ρ) = V (ρ)V (ρ)T

ä Then:f(ρ) = Tr [V (ρ)TG(ρ)V (ρ)]

= Tr [G(ρ)V (ρ)V (ρ)T ]

= Tr [G(ρ)P (ρ)].

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ä Recall [e.g.Kato ’65] that:

P (ρ) =−1

2πi

∫Γ

(G(ρ)− zI)−1 dz

Γ is a smooth curve containing the p eivenvalues of interest

ä Hence: f(ρ) =−1

2πiTr∫

ΓG(ρ)(G(ρ)− zI)−1 dz = ...

=−1

2πiTr∫

Γz(G(ρ)− zI)−1 dz

ä With this, can prove :

1. f is a non-increasing function of ρ;2. f(ρ) = 0 iff ρ = ρ∗;3. f ′(ρ) = −Tr [V (ρ)TBV (ρ)]

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Can now use Newton’s method.

ρnew = ρ−Tr [V (ρ)T(A− ρB)V (ρ)]

−Tr [V (ρ)TBV (ρ)]=

Tr [V (ρ)TAV (ρ)]

Tr [V (ρ)TBV (ρ)]

ä Newton’s method to find the zero of f ≡ a fixed point

iteration with g(ρ) =Tr [V T(ρ)AV (ρ)]

Tr [V T(ρ)BV (ρ)],

ä Idea: Compute V (ρ) by a Lanczos-type procedure

ä Note: Standard problem - [not generalized]→ inexpensive!

ä See T. Ngo, M. Bellalij, and Y.S. 2010 for details

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GRAPH-BASED TECHNIQUES

Graph-based methods

ä Start with a graph of data. e.g.: graphof k nearest neighbors (k-NN graph)Want: Perform a projection which pre-

serves the graph in some sense

ä Define a graph Laplacean:

L = D −W

x

xj

i

yi

yj

e.g.,: wij =

{1 if j ∈ Adj(i)0 else D = diag

dii =∑j 6=i

wij

with Adj(i) = neighborhood of i (excluding i)

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Example: The Laplacean eigenmaps approach

Laplacean Eigenmaps [Belkin-Niyogi ’01] *minimizes*

F(Y ) =n∑

i,j=1

wij‖yi − yj‖2 subject to Y DY > = I

Motivation: if ‖xi − xj‖ is small(orig. data), we want ‖yi − yj‖ to bealso small (low-Dim. data)ä Original data used indirectlythrough its graphä Leads to n× n sparse eigenvalueproblem [In ‘sample’ space]

x

xj

i

yi

yj

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Locally Linear Embedding (Roweis-Saul-00)

ä Very similar to Eigenmaps - but ...

ä ... Graph Laplacean is replaced by an ‘affinity’ graph

Graph: Each xi written as a convexcombination of its k nearest neighbors:xi ≈ Σwijxj,

∑j∈Adj(i)wij = 1

ä Optimal weights computed(’local calculation’) by minimizing‖xi − Σwijxj‖ for i = 1, · · · , n

x

xj

i

ä Mapped data (Y ) computed by minimizing∑‖yi − Σwijyj‖2

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Implicit vs explicit mappings

ä In PCA the mapping Φ from high-dimensional space (Rm)to low-dimensional space (Rd) is explicitly known:

y = Φ(x) ≡ V Tx

ä In Eigenmaps and LLE we only know

yi = φ(xi), i = 1, · · · , n

ä Mapping φ is complex, i.e.,

ä Difficult to get φ(x) for an arbitrary x not in the sample.

ä Inconvenient for classification

ä “The out-of-sample extension” problem

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ONPP (Kokiopoulou and YS ’05)

ä Orthogonal Neighborhood Preserving Projections

ä A linear (orthogonoal) version of LLE obtained by writing Yin the form Y = V >X

ä Same graph as LLE. Objective: preserve the affinity graph(as in LEE) *but* with the constraint Y = V >X

ä Problem solved to obtain mapping:

minV

Tr[V >X(I −W>)(I −W )X>V

]s.t. V TV = I

ä In LLE replace V >X by Y

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Face Recognition – background

Problem: We are given a database of images: [arrays of pixelvalues]. And a test (new) image.

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Face Recognition – background

Problem: We are given a database of images: [arrays of pixelvalues]. And a test (new) image.

↖ ↑ ↗

Question: Does this new image correspond to one of thosein the database?

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Example: Eigenfaces [Turk-Pentland, ’91]

ä Idea identical with the one we saw for digits:

– Consider each picture as a (1-D) column of all pixels– Put together into an arrayA of size # pixels×# images.

. . . =⇒ . . .

︸ ︷︷ ︸A

– Do an SVD ofA and perform comparison with any test imagein low-dim. space

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Graph-based methods in a supervised setting

Graph-based methods can be adapted to supervised mode.Idea: Build G so that nodes in the same class are neighbors.If c = # classes, G consists of c cliques.

ä Weight matrixW =block-diagonalä Note: rank(W ) = n− c.ä As before, graph Laplacean:

Lc = D −WW =

W1

W2. . .

Wc

ä Can be used for ONPP and other graph based methods

ä Improvement: add repulsion Laplacean [Kokiopoulou, YS09]

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Class 1 Class 2

Class 3

Leads to eigenvalue problemwith matrix:

Lc − ρLR

• Lc = class-Laplacean,• LR = repulsion Laplacean,• ρ = parameter

Test: ORL 40 subjects, 10 sample images each – example:

# of pixels : 112× 92; TOT. # images : 400

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0.98ORL −− TrainPer−Class=5

onpppcaolpp−Rlaplacefisheronpp−Rolpp

ä Observation: some values of ρ yield better results thanusing the optimum ρ obtained from maximizing trace ratio

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LINEAR ALGEBRA METHODS: EXAMPLES

IR: Use of the Lanczos algorithm (J. Chen, YS ’09)

ä Lanczos algorithm = Projection method on Krylov subspaceSpan{v,Av, · · · , Am−1v}

ä Can get singular vectors with Lanczos, & use them in LSI

ä Better: Use the Lanczos vectors directly for the projection

ä K. Blom and A. Ruhe [SIMAX, vol. 26, 2005] perform aLanczos run for each query [expensive].

ä Proposed: One Lanczos run- random initial vector. Thenuse Lanczos vectors in place of singular vectors.

ä In short: Results comparable to those of SVD at a muchlower cost.

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Tests: IR

Informationretrievaldatasets

# Terms # Docs # queries sparsityMED 7,014 1,033 30 0.735CRAN 3,763 1,398 225 1.412

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ngtim

es

Med dataset.

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Average retrieval precision

Med dataset

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Retrieval precision comparisons

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Updating the SVD (E. Vecharynski and YS’13)

ä In applications, data matrix X often updated

ä Example: Information Retrieval (IR), can add documents,add terms, change weights, ..

Problem

Given the partial SVD of X, how to get a partial SVD of Xnew

ä Will illustrate only with update of the form Xnew = [X,D](documents added in IR)

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Updating the SVD: Zha-Simon algorithm

ä Assume A≈UkΣkVTk and AD = [A,D] , D ∈ Rm×p

ä Compute Dk = (I − UkUTk )D and its QR factorization:

[Up, R] = qr(Dk, 0), R ∈ Rp×p, Up ∈ Rm×p

Note: AD≈[Uk, Up]HD

[Vk 00 Ip

]T; HD ≡

[Σk U

TkD

0 R

]ä Zha–Simon (’99): Compute the SVD of HD & get approxi-mate SVD from above equation

ä It turns out this is a Rayleigh-Ritz projection method for theSVD [E. Vecharynski & YS 2013]

ä Can show optimality properties as a result

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Updating the SVD

ä When the number of updates is large this becomes costly.

ä Idea: Replace Up by a low dimensional approximation:

ä Use U of the form U = [Uk, Zl] instead of U = [Uk, Up]

ä Zl must capture the range of Dk = (I − UkUTk )D

ä Simplest idea : best rank–l approximation using the SVD.

ä Can also use Lanczos vectors from the Golub-Kahan-Lanczosalgorithm.

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An example

ä LSI - with MEDLINE collection: m = 7, 014 (terms), n =1, 033 (docs), k = 75 (dimension), t = 533 (initial # docs),nq = 30 (queries)

ä Adding blocks of 25 docs at a time

ä The number of singular triplets of (I−UkUTk )D using SVD

projection (“SV”) is 2.

ä For GKL approach (“GKL”) 3 GKL vectors are used

ä These two methods are compared to Zha-Simon (“ZS”).

ä We show average precision then time

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Avera

ge p

recis

ion

Number of documents

Retrieval accuracy, p = 25

ZS

SV

GKL

ä Experiments show: gain in accuracy is rather consistent

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Tim

e (

se

c.)

Updating time, p = 25

Number of documents

ZS

SV

GKL

ä Times can be significantly better for large sets

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APPLICATION TO MATERIALS

Data mining for materials: Materials Informatics

ä Huge potential in exploiting two trends:

1 Improvements in efficiency and capabilities in computa-tional methods for materials

2 Recent progress in data mining techniques

ä Current practice: “One student, one alloy, one PhD” [seespecial MRS issue on materials informatics]→ Slow ..

ä Data Mining: can help speed-up process, e.g., by exploringin smarter ways

Issue 1: Who will do the work? Few researchers are familiarwith both worlds

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Issue 2: databases, and more generally sharing, not too com-mon in materials

The inherently fragmented and multidisciplinary nature ofthe materials community poses barriers to establishingthe required networks for sharing results and information.One of the largest challenges will be encouraging scien-tists to think of themselves not as individual researchersbut as part of a powerful network collectively analyzingand using data generated by the larger community.These barriers must be overcome.

NSTC report to the White House, June 2011.

ä Materials genome initiative [NSF]

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Unsupervised learning

ä 1970s: Unsupervisedlearning “by hand”: Findcoordinates that will clus-ter materials according tostructureä 2-D projection fromphysical knowledgeä ‘Anomaly Detection’:helped find that compoundCu F does not exist

see: J. R. Chelikowsky, J. C. Phillips, Phys Rev. B 19 (1978).

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Question: Can modern data mining achieve a similar dia-grammatic separation of structures?

ä Should use only information from the two constituent atoms

ä Experiment: 67 binary ‘octets’.

ä Use PCA – exploit only data from 2 constituent atoms:

1. Number of valence electrons;

2. Ionization energies of the s-states of the ion core;

3. Ionization energies of the p-states of the ion core;

4. Radii for the s-states as determined from model potentials;

5. Radii for the p-states as determined from model potentials.

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ä Result:

−2 −1 0 1 2 3−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

BN

SiC

MgS

ZnO

MgSe

CuCl

MgTe

CuBr

CuICdTe

AgI

ZWDRZWWR

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Supervised learning: classification

Problem: classify an unknown binary compound into itscrystal structure class

ä 55 compounds, 6 crystal structure classesä “leave-one-out” experiment

Case 1: Use features 1:5 for atom A and 2:5 for atom B. Noscaling is applied.

Case 2: Features 2:5 from each atom + scale features 2 to 4by square root of # valence electrons (feature 1)

Case 3: Features 1:5 for atom A and 2:5 for atom B. Scalefeatures 2 and 3 by square root of # valence electrons.

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Three methods tested

1. PCA classification. Project and do identification in space ofreduced dimension (Euclidean distance in low-dim space).

2. KNN K-nearest neighbor classification –

3. Orthogonal Neighborhood Preserving Projection (ONPP) - agraph based method - [see Kokiopoulou, YS, 2005]

Recognition rates for 3different methods usingdifferent features

Case KNN ONPP PCACase 1 0.909 0.945 0.945Case 2 0.945 0.945 1.000Case 3 0.945 0.945 0.982

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Recent work

ä Some data is becoming available

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Recent work

ä Exploit Band-sctructues - in thesame way we useimages..ä For now we doclustering.

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ä Work in progressä 3-way clusteringobtained with dim. re-duction + k-means→ä Working on unrav-eling the info & explor-ing classification withthe data

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Conclusion

ä Many, interesting new matrix problems in areas that involvethe effective mining of data

ä Among the most pressing issues is that of reducing compu-tational cost - [SVD, SDP, ..., too costly]

ä Many online resources available

ä Huge potential in areas like materials science though inertiahas to be overcome

ä On the + side: materials genome project is starting to ener-gize the field

ä To a researcher in computational linear algebra : big tide ofchange on types or problems, algorithms, frameworks, culture,...

ä But change should be welcome

ä In the words of “Who Moved My Cheese?” [ Spencer John-son, 2002]:

“If you do not change, you can become extinct !”

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“If you do not change, you can become extinct !”

“The quicker you let go of old cheese, the sooner you find newcheese.”

Thank you !

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